Abstract
In this study, the effect of support values on
the natural frequency and critical flow velocity
of a straight pipe conveying laminar flowing
fluid is studied. The aim of this work is
deriving a new analytical model to perform a
general study to investigate the dynamic
behavior of a pipe under general boundary
conditions by considering the supports as
compliant material with linear and rotational
springs. This model describes both the
classical (simply support, free, built, guide)
and the restrained boundary condition and it is
not required to derive a new frequency
equation if the boundary conditions is changed
,also the result will be near to reality by
knowing the physical parameters for the
compliant material and the pipe.

Keywords:Pipe conveying fluid, Natural
frequency,
support

Critical

velocity,

and

Elastic

1.Introduction
The study of the dynamic behavior of a fluid
conveying pipe, started in 1950, despite the
great importance of this subject in pumps, heat
exchanger, discharge lines, marines risers,
etc.., the first observation of this phenomena
was made by Ashley and Havilland [1], when
examining the above ground Trans-Arabian oil
pipe line. They considered the problem as a
simply supported pipe.
The free vibrations of pipes conveying fluid
was studied by Huang [2], taking into account,
the effects of rotary inertia on both the fluids
and the pipes, the shear deformation of the
pipes and the leteral inertia force due to the
moving fluids. The theory of dynamics
of pipes containing flowing fluid has been of a
great interest for engineers in various fields.
The results showed that the fluid forces and the
pipe forced will interact with each other. Singh
and Mallik [3] investigated the effect of
harmonic fluid on stability using bolotin,s
concept, it was concluded that, the same
regions of instability also exist in continuous

NUCEJ Vol.16 No1

Thaier J.Ntayeesh
Mechanical Eng. Dept.
University of Baghdad

pipe when it was parametrically exited and the
phenomenon was like that of a beam subjected
to an axial harmonic forces. The mass ratio did
not have any significant effect on the regions
of
instability within the range of parametric
values considered. With the increase in the
pressure and the velocity of the fluid, the
instability regions became wider and were
shifted to lower frequencies, damping in the
pipe reduced the extent of the instability
regions and a finite value of the excitation
parameters was required to start the instability.
Abraham [4] studied the vibration and stability
of straight pipe systems conveying fluid, either
steady or fluctuating flow. The supports
considered are different in type and positions,
this work perfferent a general study to
investigate the dynamic behavior for a pipe
with N-spans and general boundary conditions
and that by considering the supports as
compliant boundary material with linear and
rotational springs and dampers. It was
concluded that the support position and values
had a significant on dynamic characteristics of
the pipe.
Dian etal [5] analyzed the free lateral vibration
of thin annular with variable thickness and
circular plates. Study was adopted the finite
element method to obtain the natural
frequencies and mode shapes of the
axisymmetric and non axisymmetric thin
annular. The results showed that the finite
element method was an efficient and
convenient tool for analyzing the lateral
vibration of annular and circular composite
plates with variable thickness.
The effect of induced vibration of a simply
supported pipe conveying fluid with a
restriction, investigated theoretically and
experimentally by
Alaa [6], where transfer
matrix approach was implemented to described
the dynamic response of a pipe conveying fluid
and a numerical technique for solving twodimensional incompressible steady viscous
flow
for
the
rang
of
Reynolds
number(5<Re<1000). It was concluded that the
fluid flow through a pipe with restriction

Al-Hilli, Ntayeesh

9

The discrtized dynamical equations using
spatial finite-difference schemes in the case of
steady flow and pulsatile flow.1
Free Vibration Characteristics
10
.For a single-span pipe
conveying fluid.
4. subharmonic or to chaotic states. showed that the outer pipe length is a
more important design factor than gravity and
friction.
Wang and Bloom [7] studied the static and
dynamic instabilities of submerged and
inclined concentric pipes conveying fluid
mathematically.
Derivation of the equation of motion for stright
pipe with steady flow are available in the
literature Ref.(1)
Stiffness term
Curvature term
Coriolis force term
Inertia force term
NUCEJ Vol.
3.
7. Neglecting the effect of gravity. All motion considered small. The pipe is inextensible.
5.
it will be considered that the pipe is supported
at the two end points.section of the pipe. The
convergence study is based on the numerical
values. Neglecting the material damping.
Shintaro
and
Masaki
[8]
studied
experimentally the three-dimension dynamics
of hanging tube conveying fluid with varying
the length of the tube. In the numerical examples.the equation based on
beabeam theory is given by. Neglecting the shear deformation and rotary
inertia.affected the dynamic behavior of the pipe in
addition to the flow field structure due to
induced vibration. The pipe considered to be horizontal.
2.
In the present study. the free vibration of
elastically supported pipe conveying fluid is
analyzed under general boundary conditions by
considering the supports as compliant material
with linear and rotational springs.16 No.Ref [11]…………….
6. the first
three eigenvalues of the Timoshenko beam are
calculated for various values of stiffness of
translational and rotational springs
2.[10]. were found
depending on boundary conditions. where the parameters
and are taken to have the same values at all
the supports
Figure (1) Considered Timoshenko beams with
(a) the first type and (b) the second type of
translational and rotational springs
The following assumptions are considered in
the analysis of the system under consideration
[9]:
1. plane or rotating pendulum. The result was
obtained.Theoretical approach
Consider a straight uniform single-span pipe
conveying fluid of length L where and are
the translational and rotational spring constant.
(a)
denoted by
and
..
1
…. Instabilities to static
buckling. Neglecting the velocity distribution through
the cross.

The equation of motion Eq.
To solve the problem of free vibration for a
structure.(1) can be
written in the following non-dimensional
form:
5
2
Substituting Eq.Evaluating Natural Frequency
The flexible-flexible boundary condition may
be written as:-
NUCEJ Vol. (5) gives:
where
the
boundary
1-at x=0.
.
where
+
+
+
=0
: dimensionless
rotational stiffness at x=0
: Non-dimensional mass ratio. The
parameter of this material will be represented
by linear and rotational spring. To describe the
classical boundary conditions impedance
values are taken to be zero or infinity values. Substituting Eq.
2-at x=0.
The general solution to Eq. The method of solution consists of
formulating the support condition of a pipe in
terms of the compliant boundary material.
where
: dimensionless
rotational stiffness at x=
4-at x=l. (3) into
conditions Eq.The Coriolis force is a result of the
rotation of the system element due to the
system lateral motion.
. (2) is given
by:
.
(3) into Eq.16 No1
where
:
dimensionless
longitudinal stiffness at x=
These four equations can be written in matrix
form as follows:
Al-Hilli. a relationship is obtained
between the wave numbers
and the
where
: dimensionless
longitudinal stiffness at x=0
3-at x=l.1. Ntayeesh
11
.
… 3
where Cj is amplitudes of vibrations and
is the wave numbers.
+
+
+
=0
4
From these relationships four wave numbers
can be determined as functions of Ω and the
pipe parameters.
(
.
. (2). since each point in
the span rotates with angular velocity [12].
eigenvalues Ω.
. first its boundary conditions must be
known.
: Non-dimensional fluid pressure.
.
2.

When the flow velocity
is equal to the critical velocity the pipe bows
out and buckles. so deleting time dependent term from
equation (2) yields:
…
This matrix can be written as follows:
6
whose its solution takes the form
where
=1.
2.
The non-dimension natural frequency is
evaluated by setting term
equal to
zero
For non trivial solution:
i.
where
:dimensionless
longitudinal stiffness at x=0
3.
the
Therefore the frequency can be written
as.
2-at x=0.
dimensionless
.
The solution is done by trial and error
procedure.Therefore
the
mechanism
underlying
instability may be illustrated by static method
[13].at x=l.
A MATLAB computer program was built for
this purpose.
where
:
rotational stiffness at x=0.e.
where
and
are constant and can be evaluated by using the
boundary conditions as following:
1. the value of Ω that makes the
determinant vanish can be found which will
represent the non dimensional natural
frequency. the flow velocity in this case is called
critical flow velocity.
.at x=0. because the forces required to
make the fluid deform to the pipe curvature are
greater than the stiffness of the pipe.2.
This is determinant function to
:dimensionless
rotational stiffness at x=
4.but
function
to
therefore
determinant can be written as.
.
.1
where
:dimensionless
longitudinal stiffness at x=
These four equations can be written in matrix
form as follows:
This matrix can be written as follows:
where
=1.
. Evaluating Critical Velocity
If the natural frequencies of the pipe reach
to zero.4
Trial and error procedure is used to find the
value of
that makes the determinant
Free Vibration Characteristics
12
.at x=l.
NUCEJ Vol.3.2.1.
where
.16 No.
.4
.3.

000
38.[14]
present
work
39.579
2.
3.
and the results are given in tables 1 and 2.[14]
9. the
first three eigen values of the pipe at
with the two
types of translational and rotational springs as
shown in Figure (1) are calculated and three
dimensional plots in Figures. Results and Discussions.
Therefore.864
9. The stiffness parameters
and
are taken as having the values at all the
supports
denoted
by
.
It is possible to simulate infinite support
stiffness by setting the translational or
rotational stiffness coefficient equal to
at
all the supports for comparing the obtained
results with the existing results of the
classically
supported
pipe. From
the non-dimensional critical velocity
is
obtained.
Also.
.347
8.
.
and
by
. Effect of the Numerical Value of
Support Stiffness.652
7.583
35. Ntayeesh
13
.
for the pipe with the first type of the springs.865
9.898
present
work
39.400
37.013
38.424
37.579
2.354
Al-Hilli.474
39.Vanish.
3.3460
8.075
88.728
86.899
Ref.
In order to investigate the influence of stiffness
of the supports on the free vibration
characteristics of a pipe conveying fluid.
(6) and (7) respectively to illustrate how the
frequency parameters change with the spring
constants. by setting the translational and rotational
stiffness coefficients equal to zero at all the
supports.340
87.822
88.478
39.1.1
1
1.
f
or the beam with the second type of the
springs.652
7.16 No1
present
work
9. (3).865
84.569
34. (4).5
2
3
NUCEJ Vol. (2).
comparative study of the pinned-pinned
and
clamped-clamped
pipe with the classical
solutions given in the Ref.
u
0.
A MATLAB computer program was built for
this purpose.946
Ref. [14] is carried out. (5).
. a completely free pipe situation can
be obtained.
Table (1) Comparison study of the first two dimensionless
frequencies parameter
of the pinned-pinned pipe conveying
fluid for various values of u at
.

The effect of the fluid flow
velocity and mass ratio will be discussed. Ntayeesh
15
.
there
is
no
remarkable change in the frequency
parameters. the value
of transverse displacement decreasing when
the value of linear stiffness is increasing.87 but. Effect of Fluid Velocity and Mass Ratio. namely.226 to 21. For instance..096 to 7.351 . this
tends to increasing in natural frequency. This force
is essentially a negative damping mechanism
Al-Hilli. the frequency
changes from 4.
.
. when the
parameters
and
are both changed from
to
Figure (6) The second frequency parameter of the
pipe with
. (2) to (7).096 to 39. The reason of this behavior. Also.
and mass ratio) have direct effects on the
dynamic characteristics of the system under
consideration. value of
and the parameter
to
is changed from
. then the case will be a normal
beam system and when the flow velocity equal
the critical velocity the pipe bows out and
buckles. and is always in
phase with the velocity of the pipe. the first frequency parameter
changes from 1..
When the values of
and
are greater than
and
.2. this
tends to small increasing in natural
frequency.16 No1
and the parameter
3. for the
pipe with the first type of the springs.
Figure (7) The third frequency parameter of the
pipe with
.
Mathematically the buckling instability
arises from the mixed derivative term(Coriolis)
in equation (4) which represents a
forces imposed on the pipe by the flowing fluid
that
is always 90o of phase with the
displacement of the pipe. when the
parameters
and
are
taken
as
.
The fluid parameters (velocity.
becomes 17 times greater in this change. pressure. If the velocity of the flow in the
pipe equal zero. But
the value of slope is small decreasing when the
value of rotational stiffness increasing. because the forces required to make
the fluid deform to the pipe curvature is greater
than the stiffness of the pipe. the frequency parameter
changes from 4.is changed from
parameter
to
. when the
spring parameter is taken constant. This situation can be observed
from the flat area of Figures. then. it
is evident from the obtained values of
frequency parameters that.
In general the natural frequencies for
steady flow decrease with increasing the fluid
flow velocity.243. the pipe can be
considered as a pipe fixed at the both ends. in Figure (3).
respectively in the considered change.
Increment in the values of parameters and
are more effective on the first frequency
parameter of the pipe than the second and third
frequency parameters.For example. while the
parameter
is taken as
NUCEJ Vol. Because the
value of natural frequency in the first mode is
small and any increasing in
and
strong
effect in the value of natural frequency but the
natural frequency in the second and third
modes is high and the increasing in
and
small effect in the value of natural frequency.
It is seen from the figures that
translational springs are much more effective
on the frequency parameters than rotational
springs.
and
increase
approximately 14 and 5 times. On
the other hand.

such as
cantilever pipe. there is
no difference in the frequency parameter for
any value of . vibration. 12and 13) show the variation
of the frequency parameter
with flow
velocity parameter
the mass ratio
Figure (8) Variation of
with
for various values of
for Clamped. The more
effect happen when the value of stiffness is
small and this effect decreasing when the value
of stiffness is increasing.1
Figure (9) Variation of
with for various values of
for pinned.16 No. For intermediate values of .
Free Vibration Characteristics
16
. For
and for
.
Many studies indicate that the natural
frequencies of a pipe with both ends stationary. are strongly affected by the
mass ratio [15].which extracts energy from the fluid flow and
inputs energy into the bending pipe to
encourage initially.
Figures (8) and (9) shows the variation of the
frequency parameter
with flow velocity
parameter
for different values of the mass
ratio for the clamped –clamped and pinnedpinned boundary condition respectively. and ultimately
buckling [15].
such as the clamped-clamped or clampedpinned ends.
Figures (10. are nearly independent of the
mass ratio while the natural frequencies of
pipes with one end free to move.pinned pipe. 11.
NUCEJ Vol. The effect
of mass ratio on the natural frequency
depended on the value of rotational and
translational impedance of support. for the pinned-pinned and clampedclamped cases.
There is very good agreement in the values of
with those obtained by Païdoussis and
Issid [16]. so less effect happens
at the value of stiffness is large.
there is a slight decrease in the frequency
parameter for increasing values of
in
classical boundary condition [16].Clamped pipe
for different values of
for four types of flexible
support.

4-The values of linear and rotational
impedance are more effective on the first
frequency parameter of the pipe than the
second and third frequency parameters. =100
Figure (13) Variation of
with for various
values of at =5. there is no difference in the frequency
parameter for any value of mass ratio.16 No1
Following the main summarized
conclusions raised by this research:
1-The technique used for modeling the
compliant boundary material in terms of linear
and rotational impedance allows the designer
to describe both the classical and restrained
boundary conditions. The reason of this behavior. =180 .
Al-Hilli.When flow velocity equal zero or critical
value.
7. =175 .
But the value of slope is small decreasing
when the value of rotational stiffness
increasing. there is
decrease in the natural frequency parameter
with increasing values of mass ratio. =520. =130 .
5-When the values of linear and rotational
impedance are greater than
. =450. this tends to small increasing in
natural frequency. =380
3.
this tends to increasing in natural frequency.
2-The natural frequency of a pipe increases
with the increasing of the linear and rotational
impedance. for
intermediate values of flow velocity. =250
NUCEJ Vol. Ntayeesh
17
. there is no
remarkable change in the frequency
parameters. =10. =170.
3-The linear impedance is much more effective
on the frequency parameters than rotational
impedance. the
value of transverse displacement decreasing
when the value of linear stiffness is increasing. =10 . =90
[[[[[
Figure (12) Variation of
with for various
values of
at =750. Conclusions
Figure (11) Variation of
with for various
values of
at =110.Figure (10) Variation of
with for various
values of at =100.
6-The natural frequency of a pipe increases
with the increasing of the linear and rotational
intermediate impedance.